Method for determining anisotropic resistivity and dip angle in an earth formation

ABSTRACT

A method for determining the anisotropic resistivity properties of a subterranean formation traversed by a borehole utilizing a multi-component induction logging tool. The method utilizes the measured phase and attenuation signals induced by eddy currents in the formation to determine the azimuthal angle and to create a second set of signals based on the measured signals and the azimuthal angle. The anisotropic resistive properties, as well as the dip angle, are then simultaneously derived from the second set of signals by means of minimizing error functions within an inversion model.

This application claims the benefit of U.S. Provisional Application No.60,336,996 filed Dec. 3, 2001, the entire disclosure of which is herebyincorporated by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention is related to the field of electromagnetic loggingof earth formations penetrated by well borehole. More specifically, theinvention is related to a method for determining the anisotropicresistivity properties of the earth formation and the dip angle of theborehole in the earth formation.

2. Description of the Related Art

The basic techniques of electromagnetic or induction logging instrumentsare well known in the art. A sonde, having at least one transmitter coiland at least one receiver coil, is positioned in a well borehole eitheron the end of a wireline or as part of a logging while drilling (“LWD”).The axis of the coils is essentially co-linear with the axis of thesonde and borehole. An oscillating signal is transmitted throughtransmitter coil, which creates a magnetic field in the formation. Eddycurrents are induced in the earth formation by the magnetic field,modifying the field characteristics. The magnetic field flows in groundloops essentially perpendicular to the tool axis and is picked up by thereceiver coil. The magnetic field induces a voltage in the receiver coilrelated to the magnitude of the earth formation eddy currents. Thevoltage signals are directly related to the conductivity of the earthformation, and thereby conversely the formation resistivity. Formationresistivity is of interest in that one may use it to infer the fluidcontent of the earth formation. Hydrocarbons in the formation, i.e. oiland gas, have a higher resistivity (and lower conductivity) than wateror brine.

However, the formation is often not homogeneous in nature. Insedimentary strata, electric current flows more easily in a directionparallel to the strata or bedding planes as opposed to a perpendiculardirection. One reason is that mineral crystals having an elongatedshape, such as kaolin or mica, orient themselves parallel to the planeof sedimentation. As a result, an earth formation may posses differingresistivity/conductivity characteristics in the horizontal versusvertical direction. This is generally referred to as formationmicroscopic anisotropy and is a common occurrence in minerals such asshales. The sedimentary layers are often formed as a series ofconductive and non-conductive layers. The induction tool response tothis type of formation is a function of the conductive layers where thelayers are parallel to the flow of the formation eddy currents. Theresistivity of the non-conductive layers is represents a small portionof the received signal and the induction tool responds in a manner.However, as noted above, it is the areas of non-conductivity (highresistivity) that are typically of the greatest interest when exploringfor hydrocarbons. Thus, conventional induction techniques may overlookareas of interest.

The resistivity of such a layered formation in a direction generallyparallel to the bedding planes is referred to as the transverse orhorizontal resistivity R_(h) and its inverse, horizontal conductivityσ_(h). The resistivity of the formation in a directive perpendicular tothe bedding planes is referred to as the longitudinal or verticalresistivity R_(v), with its inverse vertical conductivity σ_(v). Theanisotropy coefficient, by definition is: $\begin{matrix}{\lambda = {\sqrt{R_{H}/R_{V}} = {\sqrt{\sigma_{v}/\sigma_{h}} = \frac{1}{\alpha}}}} & \lbrack 1\rbrack\end{matrix}$

Subterranean formations are often made up of a series of relatively thinbeds having differing lithological characteristics and resistivities.When the thin individual layers cannot be delineated or resolved by thelogging tool, the logging tool responds to the formation as if it weremacroscopically anisotropic formation, ignoring the thin layers.

Where the borehole is substantially perpendicular to the formationbedding planes, the induction tool responds primarily to the horizontalcomponents of the formation resistivity. When the borehole intersectsthe bedding planes at an angle, often referred to as a deviatedborehole, the tool will respond to components of both the vertical andhorizontal resistivity. With the increase in directional and horizontaldrilling, the angle of incidence to the bedding planes can approach 90°.In such instances, the vertical resistivity predominates the toolresponse. It will be appreciated that since most exploratory wells aredrilled vertical to the bedding planes, it may be difficult to correlateinduction logging data obtained in highly deviated boreholes with knownlogging data obtained in vertical holes. This could result in erroneousestimates of formation producibility if the anisotropic effect is notaddressed.

A number of techniques and apparatus have been developed to measureformation anisotropy. These techniques have included providing theinduction tool with additional transmitter and receiver coils, where theaxes of the additional coils are perpendicular to the axes of theconventional transmitter and receiver coils. An example of this type oftool might include U.S. Pat. No. 3,808,520 to Runge, which proposedthree mutually orthogonal receiver coils and a single transmitter coil.Other apparatus include the multiple orthogonal transmitter and receivercoils disclosed in U.S. Pat. No. 5,999,883 to Gupta et al. Still othertechniques have utilized multiple axial dipole receiving antennae and asingle multi-frequency transmitter, or multiple axial transmitters suchas those described in U.S. Pat. No. 5,656,930 to Hagiwara and U.S. Pat.No. 6,218,841 to Wu.

SUMMARY OF THE INVENTION

A new method is provided for determining the anisotropic properties of asubterranean earth formation. The present invention is directed to amethod for determining the anisotropic properties of an earth formationutilizing a multi-component induction. Specifically, the presentinvention contemplates a method for inverting the multi-componentinduction tool responses to determine anisotropic resistivity of ananisotropic and/or homogeneous formation and determine the tool'sorientation with respect to the formation anisotropic directionutilizing both the resistive (R) and reactive (X) portions of thesignals from a combination of tool responses.

In a preferred implementation, an induction logging tool, havingmultiple mutually orthogonal transmitter coils and receiver coils, ispositioned in a borehole and activated. Power is applied to thetransmitter coils to induce eddy currents in the formation. These eddycurrents then induce currents in the receiver coils. These are processedto generate a preliminary phase shift derived resistivity andattenuation derived resistivity. This information is then compared witha predetermined model that relates phase shift derived resistivity andattenuation derived resistivity, horizontal resistivity, verticalresistivity and the anisotropy coefficient. Utilizing an inversiontechnique based on the preexisting model, the horizontal resistivity andvertical resistivity for a formation, as well as anisotropy coefficientand deviation angle relative to the formation, may be readily determinedfrom the logging data.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present invention may be had byreferencing detailed description together with the Figures, in which:

FIG. 1 is simplified depiction of a logging tool that may be used topractice the present invention deployed in an earth borehole;

FIG. 2A depicts the orientation of the tool transmitters and receivers;

FIG. 2B depicts the relationship between the borehole, formation andtool coordinate systems;

FIG. 3A is a nomograph depicting a means for determining R_(H) as afunction of R_(ll);

FIG. 3B is a nomograph depicting a means for determining R_(H) as afunction of R_(ll) and X_(ll);

FIG. 3C is a nomograph depicting a means for determining R_(V)/R_(H) asa function of R_(tt);

FIG. 3D is a nomograph depicting a means for determining R_(V)/R_(H) asa function of the ratio R_(tt)/R_(ll);

FIG. 4A is a nomograph depicting a means for determining R_(H) and R_(V)as a function of R_(tt) and X_(tt);

FIG. 4B is a nomograph depicting a means for determining R_(V)/R_(H) asa function of the ratio R_(tt)/X_(tt);

FIG. 5A is a nomograph depicting a means for determining R_(H) as afunction R_(tt);

FIG. 5B is a nomograph depicting a means for determining R_(H) as afunction of R_(tt) and X_(tt);

FIG. 5C is a nomograph depicting a means for determining R_(V)/R_(H) asa function of R_(ll);

FIG. 5D is a nomograph depicting a means for determining R_(V)/R_(H) asa function of R_(uu);

FIG. 5E is a nomograph depicting a means for determining R_(V)/R_(H) asa function of the ratio of R_(ll)/R_(tt);

FIG. 5F is a nomograph depicting a means for determining R_(V)/R_(H) asa function of the ratio of R_(uu)/R_(tt);

FIG. 6A is a nomograph depicting a means for determining R_(H) and R_(V)as a function of R_(ll) and X_(ll);

FIG. 6B is a nomograph depicting a means for determining R_(V)/R_(H) asa function of the ratio of X_(ll)/R_(ll);

FIG. 6C is a nomograph depicting a means for determining R_(H) and R_(V)as a function of R_(uu) and X_(uu);

FIG. 6D is a nomograph depicting a means for determining R_(V)/R_(H) asa function of the ratio of X_(uu)/R_(uu);

FIG. 7A is a nomograph depicting a means for determining R_(H) and β asa function of R_(ll) and X_(ll);

FIG. 7B is a nomograph depicting a means for determining R_(H) as afunction of the ratio of R_(ll)/X_(ll); and

FIG. 8 is a flow chart depicting the operation of the method of thepresent invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The present invention is intended to be utilized with a multi-component,i.e., multiple mutually orthogonal transmitters and receiver coils.Exemplary of this type of induction tool is that disclosed in U.S. Pat.No. 5,999,883 to Gupta et al., which is incorporated herein byreference. In FIG. 1, an induction tool 10 is disposed in a wellbore 2drilled through an earth formation 3. The earth formation 3 is shown ashaving a zone of interest 4. The tool 10 is lowered into the earthformation 3 to the zone of interest 4 on an armored, multi-conductorcable 6. The cable 6 is further part of a surface system (not shown)which might typically consist of a winch, a surface control system,including one or more surface computers, interface equipment, powersupplies and recording equipment. The surface systems of this type mightinclude a mobile truck mounted unit or a skid mounted unit for offshoreoperations. Further, the tool 10 may be transported utilizing othertechniques, such as coiled tubing having power and data communicationscapability or as part of a drilling string in a Logging While Drilling(LWD) suite of tools.

The tool 10 is comprised of three subsections including an electronicssection 14, a coil mandrel unit 8, and a receiver/processing telemetrysection 12 that is in communication with the cable 6. The coil mandrelsection 8 includes the transmitter coils for inducing an electromagneticfield in the earth formation in the zone of interest 4 upon applicationof power and the receiving coils for picking up signals created byinduced eddy currents characteristic is the zone of interest 4. Theelectronics section 14 include the signal generator and power systems toapply the current to the transmitter coils. The tool 10 is shown asbeing disposed adjacent to a zone of interest 4 that is made up of thinformation sections 4A-4E.

It should be noted that while FIG. 1 depicts the tool 10 as beinglowered in a vertical borehole 2, that current drilling techniquescommonly result in a borehole which deviates several times along itslength from the true vertical position. Accordingly, a borehole mayintersect a zone of interest at an angle and could greatly affect thetool's measurement of resistivity characteristics. The preferred methodof the present invention is designed to address this problem.

1. Relation of Tool, Borehole and Formation Coordinate Systems

A multi-component induction tool such as that disclosed in U.S. Pat. No.5,999,883 consists of at least three mutually orthogonal loop antennatransmitters (M_(l), M_(m), M_(n)) and at least three mutuallyorthogonal receiver coils whose responses are proportional to themagnetic field strength vectors (H_(l), H_(m), H_(n)), where l, m, and ndenote a common coordinate system. It should be noted that the in-phaseR-signal is proportional to the imaginary part of the H field. The threetransmitters are set at the same position longitudinally along the tool10 axis, which coincides with the l-axis of the coordinate system. Thethree receivers are also grouped at a common position spaced away fromthe transmitters along the l-axis. The two transverse directions arealong the m- and n-axes.

With this arrangement, there exist nine different complex voltagemeasurements proportional to the magnetic field strength vector at thereceiver loop antennas when the transmitters are activated:

(H_(ll), H_(ml), H_(nl)) from transmitter M_(l);

(H_(lm), H_(mm), H_(nm)) from transmitter M_(m); and

(H_(ln), H_(mn), H_(nn)) from transmitter M_(n).

However, as a matter of reciprocity, H_(nl)=H_(ln); H_(mn)=H_(nm); andH_(ml)=H_(lm). Accordingly, there are six independent measurements,which can be expressed as: $\begin{matrix}{H^{tool} = {\begin{bmatrix}H_{ll} & H_{l\quad m} & H_{l\quad n} \\H_{m\quad l} & H_{m\quad m} & H_{mn} \\H_{nl} & H_{n\quad m} & H_{nn}\end{bmatrix} = \begin{bmatrix}H_{ll} & H_{l\quad m} & H_{l\quad n} \\H_{l\quad m} & H_{m\quad m} & H_{mn} \\H_{l\quad n} & H_{mn} & H_{nn}\end{bmatrix}}} & \lbrack 2\rbrack\end{matrix}$

One starts with the assumption that formations 4A-4E of FIG. 1 arelayered horizontally, where the true vertical direction is z-axis. Aformation will be said to exhibit anisotropy where the resistivity inthe vertical direction is different from that in the horizontaldirection. The formation conductivity tensor is characterized by twoanisotropic conductivity values: $\begin{matrix}{\sigma = {\begin{bmatrix}\sigma_{zz} & \sigma_{xy} & \sigma_{zy} \\\sigma_{xz} & \sigma_{xx} & \sigma_{xy} \\\sigma_{yz} & \sigma_{yx} & \sigma_{yy}\end{bmatrix} = \begin{bmatrix}\sigma_{V} & 0 & 0 \\0 & \sigma_{H} & 0 \\0 & 0 & \sigma_{H}\end{bmatrix}}} & \lbrack 3\rbrack\end{matrix}$

where σ_(H) is the horizontal conductivity and σ_(V) is the verticalconductivity of the formation. The formation coordinates system in thisinstance is (z, x, y). Where the coordinate system of the tool (l, m, n)is aligned with the coordinate system of the formation (z, x, y), themagnetic field strength in the formation can be expressed as:$\begin{matrix}{H^{formation} = \begin{bmatrix}H_{zz} & H_{zx} & H_{zy} \\H_{xz} & H_{xx} & H_{xy} \\H_{yz} & H_{yx} & H_{yy}\end{bmatrix}} & \lbrack 4\rbrack\end{matrix}$

However, as noted above, the borehole is rarely vertical, meaning thecoordinate systems rarely align. This represents a deviation angle of θ.The borehole itself may be considered to have a coordinate system (l, t,u) where the u-axis coincides with the formation's y-axis. See FIG. 3.The borehole coordinate system and the formation coordinate system arerelated by a rotational operation around the y-axis by an inclinationangle θ about the y-axis: $\begin{matrix}{\begin{bmatrix}\hat{l} \\\hat{t} \\\hat{u}\end{bmatrix} = {{\begin{bmatrix}{\cos \quad \theta} & {\sin \quad \theta} & 0 \\{{- \sin}\quad \theta} & {\cos \quad \theta} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}\hat{z} \\\hat{x} \\\hat{y}\end{bmatrix}} = {{R_{y}(\theta)}\begin{bmatrix}\hat{z} \\\hat{x} \\\hat{y}\end{bmatrix}}}} & \lbrack 5\rbrack\end{matrix}$

where R_(y) is the rotational operator for angle θ.

Presuming that the longitudinal axis of the tool is aligned with theborehole coordinate system, then it may be stated that the antennacoordinate system (l, m, n) is aligned with the borehole coordinatesystem (l, t, u) and $\begin{matrix}{H^{borehole} = \begin{bmatrix}H_{ll} & H_{xy} & H_{xz} \\H_{yx} & H_{yy} & H_{yz} \\H_{zx} & H_{zy} & H_{zz}\end{bmatrix}} & \lbrack 6\rbrack\end{matrix}$

Moreover, H^(borehole) and H^(formation) are related by the rotationalfactor: $\begin{matrix}{\begin{bmatrix}H_{ll} & H_{lt} & H_{lu} \\H_{tl} & H_{tt} & H_{tu} \\H_{ul} & H_{ut} & H_{uu}\end{bmatrix} = {{{R_{y}(\theta)}\begin{bmatrix}H_{zz} & H_{zx} & H_{zy} \\H_{xz} & H_{xx} & H_{xy} \\H_{yz} & H_{yx} & H_{yy}\end{bmatrix}}{R_{y}(\theta)}^{tr}}} & \lbrack 7\rbrack\end{matrix}$

Where R_(y)(θ)^(tr) is the transposition of R_(y)(θ).

However it is rare that the coordinate system (l, m, n) of the toolantennae is aligned with the borehole coordinate system (l, t, u).Accordingly, the transverse (m, n) tool coordinates are related to theborehole coordinates (t, u) be a rotational operation about the l-axisby an azimuthal angle φ: $\begin{matrix}{\begin{bmatrix}\hat{l} \\\hat{m} \\\hat{n}\end{bmatrix} = {{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos \quad \phi} & {\sin \quad \phi} \\0 & {{- \sin}\quad \phi} & {\cos \quad \phi}\end{bmatrix}\begin{bmatrix}\hat{l} \\\hat{t} \\\hat{u}\end{bmatrix}} = {{R_{l}(\theta)}\begin{bmatrix}\hat{l} \\\hat{t} \\\hat{u}\end{bmatrix}}}} & \lbrack 8\rbrack\end{matrix}$

The tool response H^(tool) is then related to the borehole responseH^(borehole) be the operator $\begin{matrix}{\begin{bmatrix}H_{ll} & H_{lt} & H_{lu} \\H_{tl} & H_{tt} & H_{tu} \\H_{ul} & H_{ut} & H_{uu}\end{bmatrix} = {{{R_{l}(\theta)}^{tr}\begin{bmatrix}H_{ll} & H_{l\quad m} & H_{l\quad n} \\H_{m\quad l} & H_{m\quad m} & H_{mn} \\H_{nl} & H_{n\quad m} & H_{nn}\end{bmatrix}}{R_{l}(\phi)}}} & \lbrack 9\rbrack\end{matrix}$

where R_(l)(φ)^(tr) is defined as the transposition of R_(l)(φ).

2. Tool Response

Having defined the coordinate systems for the tool, borehole andformation, the tool response may now be expressed in terms of theformation coordinate system (z, x, y), where the tool (l, m, n)directions are aligned with the formation. At a tool transmitter, thefields in the formation can be described per Moran/Gianzero (J. H. Moranand S. C. Gianzero, “Effects of Formation Anisotropy on ResistivityLogging Measurements”, Geophysics (1979) 44, p. 1266) as follows:$\begin{matrix}{H_{zz} = {\frac{M_{z}}{4\pi}\left\{ {{\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\frac{z^{2}}{r^{2}}} - \left( {1 - u + u^{2}} \right)} \right\} \frac{e^{u}}{r^{3}}}} & \lbrack 10\rbrack \\{H_{zx} = {\frac{M_{z}}{4\pi}\left\{ {\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\frac{xz}{r^{2}}} \right\} \frac{e^{u}}{r^{3}}}} & \lbrack 11\rbrack \\{H_{xx} = {{\frac{M_{x}}{4\pi}\left\{ {{\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\frac{x^{2}}{r^{2}}} - \left( {1 - u + u^{2}} \right)} \right\} \frac{e^{u}}{r^{3}}} + I_{0}}} & \lbrack 12\rbrack \\{H_{yy} = {{\frac{M_{y}}{4\pi}\left\{ {- \left( {1 - u + u^{2}} \right)} \right\} \frac{e^{u}}{r^{3}}} - I_{0} + I_{1}}} & \lbrack 13\rbrack\end{matrix}$

 H _(yz)=0  [14]

H _(yx)=0  [15]

where

u=ik _(H) r  [16]

and k_(H) is the frequency of the magnetic moment induced and r is

r={square root over (x² +y ² +z ²)}; and ρ={square root over (x ² +y²)}.

The receiver coils are also presumed to be located relative to theformation coordinate system (z, x, y)=(Lcosθ, Lsinθ, 0), where θ isagain the deviation angle. The currents induced in the receiver coilsmay be expressed as: $\begin{matrix}{I_{0} = {{\frac{M_{z}}{4\pi}\left\{ {\frac{u}{r}\left( {e^{u} - e^{u\quad \beta}} \right)} \right\} \frac{1}{\rho^{2}}} = {{\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}} = {\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}{\overset{\sim}{I}}_{0}}}}} & \lbrack 17\rbrack \\{\quad {I_{1} = {{\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}{u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} = {\frac{M_{z}}{4\pi}\frac{e^{u}}{r^{3}}{\overset{\sim}{I}}_{1}}}}} & \lbrack 18\rbrack\end{matrix}$

where α is the inverse of λ and β is the anisotropy factor β={squareroot over (1+(α²−1)sin²θ)}, r=L, and ρ=Lsinθ. It should be noted that inthis instance u is a function of only the horizontal resistivity. BothI₀ and I₁ are dependent on u, β and θ. If all of the transmitters areset at equal transmission power (M₁=M_(u)=M_(t)=M₀), then the toolresponse in the borehole can be written as: $\begin{matrix}{\quad {H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left( {1 - u} \right)} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}} & \lbrack 19\rbrack \\{\quad {H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left( {1 - u + u^{2}} \right)} + {u\frac{\cos^{2}\theta}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}} & \lbrack 20\rbrack \\{\quad {H_{lt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {u\frac{\cos \quad \theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}}} & \lbrack 21\rbrack \\{H_{uu} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left( {1 - u + u^{2}} \right)} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}} & \lbrack 22\rbrack\end{matrix}$

 H _(ul)=0  [23]

H _(ut)=0  [24]

This results in three unknowns that characterize the formationsanisotropic resistivity, R_(H), R_(V), and the deviation angle θ. Notethat H_(ll) depends only on u (hence R_(H)) and β, only. The remainderof the responses, H_(tt), H_(lt), H_(uu) depend on the variables u, β,and θ.

3. Azimuthal Correction for Deviated Boreholes

In deviated boreholes, the azimuthal rotation of the tool in theborehole must be determined. In actual logging, the tool's azimuthalorientation is not known. The directions of two transversally orientedantennas do not coincide with the l- and u-axis directions. Themulti-component induction tool measures H^(tool) that is different fromH^(borehole).

In a longitudinally anisotropic formation, the orthogonality conditionholds as H_(yz)=H_(yx)=0. This implies H_(ul)=H_(ut)=0. As a result, notall six measurements of H^(tool) are independent, and (H_(lm), H_(ln),H_(mm), H_(nn), H_(mn)) must satisfy the following consistencycondition: $\begin{matrix}{H_{mn} = \frac{\left( {H_{m\quad m} - H_{nn}} \right)H_{l\quad m}H_{l\quad n}}{\left( {H_{l\quad m}^{2} - H_{l\quad n}^{2}} \right)}} & \lbrack 25\rbrack\end{matrix}$

The azimuthal angle θ is determined either from (H_(lm), H_(ln)) by,$\begin{matrix}{{\tan \quad \phi} = {- \frac{H_{l\quad n}}{H_{l\quad m}}}} & \lbrack 26\rbrack \\{{\cos \quad \phi} = \frac{H_{l\quad m}}{\sqrt{H_{l\quad m}^{2} + H_{l\quad n}^{2}}}} & \lbrack 27\rbrack \\{{\sin \quad \phi} = {- \frac{H_{l\quad n}}{\sqrt{H_{l\quad m}^{2} + H_{l\quad n}^{2}}}}} & \lbrack 28\rbrack\end{matrix}$

or from (H_(mm), H_(nn), H_(mn)) by, $\begin{matrix}{{\tan \quad 2\phi} = {{- \frac{2H_{mn}}{H_{m\quad m} - H_{nn}}} = \frac{2\tan \quad \phi}{1 - {\tan^{2}\phi}}}} & \lbrack 29\rbrack\end{matrix}$

If all of the (H_(lm), H_(ln), H_(mm), H_(nn), H_(mn)) measurements areavailable, the azimuthal angle φ may be determined by minimizing theerror, $\begin{matrix}{{error} = {{{{H_{l\quad m}\sin \quad \phi} + {H_{l\quad n}\cos \quad \phi}}}^{2} + {{{\frac{H_{m\quad m} - H_{nn}}{2}\sin \quad 2\phi} + {H_{mn}\cos \quad 2\phi}}}^{2}}} & \lbrack 30\rbrack\end{matrix}$

The H^(borehole) is calculated in terms of H^(tool) by, $\begin{matrix}{H_{lt} = \sqrt{H_{l\quad m}^{2} + H_{l\quad n}^{2}}} & \lbrack 31\rbrack \\{H_{tt} = \frac{{H_{m\quad m}H_{l\quad m}^{2}} - {H_{nn}H_{l\quad n}^{2}}}{H_{l\quad m}^{2} - H_{l\quad n}^{2}}} & \lbrack 32\rbrack \\{H_{uu} = \frac{{H_{nn}H_{l\quad m}^{2}} - {H_{m\quad m}H_{l\quad n}^{2}}}{H_{l\quad m}^{2} - H_{l\quad n}^{2}}} & \lbrack 33\rbrack\end{matrix}$

These should be used in determining the R_(H), R_(V), and θ, from themulti-component induction tool measurements H^(tool).

4. Inversion of Multi-Component Induction Resistivity Data

The present invention utilizes an inversion technique to determineanisotropic resistivity characteristics over a range of deviationangles. Where the tool is placed in vertical borehole (θ=0), there existtwo independent measurements, each independent measurement beingcomposed of an H field having both an in-phase and out-phase component.$\begin{matrix}{H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {2\left( {1 - u} \right)} \right\}}} & \lbrack 34\rbrack \\{H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {- \left( {1 - u + {u^{2}\quad \frac{\alpha^{2} + 1}{2}}} \right)} \right\}}} & \lbrack 35\rbrack\end{matrix}$

 H _(lt)=0  [36]

H _(uu) =H _(tt)  [37]

This means that H_(ll) is a function solely of u, hence the horizontalresistivity. H_(tt)=H_(uu) which means both are a function of u andanisotropy λ²=R_(H)/R_(V), i.e., a function of both the horizontal andvertical resistivity.

FIG. 3A demonstrates the relationship between the relationship betweenthe R-signal and the formation horizontal resistivity R_(H). Thisrelationship may be used to invert the conventional R signal to obtainapparent R_(H). When using both the resistive R and reactive Xcomponents of the signal measured as part of H_(ll), the graph set forthin FIG. 3B may be used to determine R_(H). This is done by minimizingthe model error as follows: $\begin{matrix}{{error} = {{H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {2\left\lbrack {1 - u} \right\rbrack} \right\}}}}^{2}} & \lbrack 38\rbrack\end{matrix}$

Once R_(H) is determined from the above, the vertical resistivity R_(V)can be determined from the H_(ll) measurement. One means of doing so isto determine the ratio R_(H)/R_(V) as a function of the R-signal as setforth in FIG. 3C. An alternative means of doing so would be based on theratio of the measured H_(tt)/H_(ll), ratio based again on the resistivecomponent of the signal R, i.e., R_(tt)/R_(ll), as demonstrated in FIG.3D.

An alternative means of determining horizontal and vertical resistivitymay be accomplished utilizing both the resistive and reactive portionsof the received signal from H_(tt). FIG. 4A is a nomograph showingdiffering ratios of anisotropic resistivity values R_(H), R_(V) (in thisinstance, as a function of λ, the square root of R_(H)/R_(V)). BothR_(H) and R_(V) can be determined simultaneously by minimizing the erroras follows: $\begin{matrix}{{error} = {{H_{tt}^{measured} + {\frac{M_{0}}{4\pi}{\frac{e^{u}}{r^{3}}\left\lbrack {1 - u + {u^{2}\quad \frac{\alpha^{2} + 1}{2}}} \right\rbrack}}}}^{2}} & \lbrack 39\rbrack\end{matrix}$

An alternative means to determine R_(H) and R_(V) is shown in nomographof FIG. 4B, which shows R_(V)/R_(H) as a function of both the R and Xsignals from H_(tt). Both R_(H) and R_(V) can be determinedsimultaneously by again minimizing Eq. 29.

When the tool records R- and X-signals for both H_(ll) and H_(tt), R_(H)and R_(V) can be determined simultaneously by minimizing the error:$\begin{matrix}{{error} = \left| {{H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {2\left\lbrack {1 - u} \right\rbrack} \right\}}}\left| {}^{2}{\quad {+ {\left. \left| {H_{tt}^{measured} + {\frac{M_{0}}{4\pi}{\frac{e^{u}}{r^{3}}\left\lbrack {1 - u + {u^{2}\quad \frac{\alpha^{2} + 1}{1}}} \right\rbrack}}} \right. \right|^{2}}}} \right.} \right.} & \lbrack 40\rbrack\end{matrix}$

As noted previously, it is rare in current drilling and logging practicethat a well is drilled vertical. At the opposite end of the spectrum isa determination of anisotropic resistivity characteristics when theborehole is essentially horizontal (θ=90°). Equations 19-24 reduce tothree independent measurements, each with a resistive and reactivecomponent: $\begin{matrix}{\quad {H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left( {1 - u} \right)} + {u\left( {1 - e^{u{({\alpha - 1})}}} \right)}} \right\}}}} & \lbrack 41\rbrack \\{\quad {H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {- \left( {1 - u + u^{2}} \right)} \right\}}}} & \lbrack 42\rbrack \\{H_{uu} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left( {1 - u + u^{2}} \right)} - {u\left( {1 - e^{u{({\alpha - 1})}}} \right)} + {u^{2}\left( {1 - {\alpha \quad e^{u{({\alpha - 1})}}}} \right)}} \right\}}} & \lbrack 43\rbrack\end{matrix}$

 H _(lt) =H _(ul) =H _(ut)=0  [44]

Herein, H_(tt) is a function solely of u, and hence is a function solelyof horizontal resistivity R_(H). H_(tt) and H_(uu) are both a functionof variables u and α, i.e., u and R_(H) and R_(V).

FIG. 5A is a nomograph that relates the R signal from H_(tt) to thehorizontal resistivity R_(H). This may be used to invert the R signal tothe apparent formation resistivity. Where R and X signals are availablefor H_(tt), they may also be used to obtain a formation horizontalresistivity for differing R_(H) as shown in nomograph 5B. This isaccomplished my minimizing the error function: $\begin{matrix}{{error} = \left| {H_{tt}^{measured} + {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {1 - u + u^{2}} \right\}}} \right|^{2}} & \lbrack 45\rbrack\end{matrix}$

Once the R_(H) is determined by either method above, the anisotropy andthe vertical resistivity can be determined from various signals based onthe H_(ll) measurement. The anisotropy R_(V)/R_(H) may be obtained fromthe R signal based on the H_(ll) measurement as shown in FIG. 5C.Alternatively, the anisotropy R_(V)/R_(H) may be obtained as a functionof the R signal from H_(uu) as shown in FIG. 5D. It should be noted thata determination using FIG. 5D is more highly dependent on R_(H) asopposed to R_(V) at high ohm-m resistivities. A more accuratedetermination of R_(V)/R_(H) may be made as a ratio of the R signalsfrom H_(uu)/H_(ll) as demonstrated in FIG. 5E. Another inversion that isuseful only at low resistivities is depicted in FIG. 5E which attemptsto derive R_(V)/R_(H) as a function of the ratio R signals received anantennae H_(uu)/H_(tt).

In the preferred method of the present invention, both the R and Xsignals are used to determine R_(H) and R_(V) simultaneously. In thenomograph of FIG. 6A, R_(V) and R_(H) are determined as a function ofthe ratio of the R/X signals at H_(ll) is obtained for differinganisotropic values. R_(H) and R_(V) may be determined by minimizing theerror function: $\begin{matrix}{{error} = \left| {H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\alpha - 1})}}} \right)}} \right\}}} \right|^{2}} & \lbrack 46\rbrack\end{matrix}$

However, the ratio of R_(V)/R_(H) may be better determined from theratio X/R signals received at H_(ll) as demonstrated in FIG. 6B. As inFIG. 6A, one may determined R_(V) and R_(H) for differing anisotropyvalues based on the ratio of the R/X signals received at H_(uu), asshown in FIG. 6C and then minimizing the following error function todetermine R_(V) and R_(H): $\begin{matrix}{{error} = \left| {H_{uu}^{measured} - {\left. {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {u\left( {1 - e^{u{({\alpha - 1})}}} \right)} + {u^{2}\left( {1 - {\alpha e}^{u{({\alpha - 1})}}} \right)}} \right\}} \right|^{2}}} \right.} & \lbrack 47\rbrack\end{matrix}$

As with the signals received at H_(ll), a better determination of theratio R_(V)/R_(H) may be made with respect to anisotropy values based onthe ratio of X/R signals received at H_(uu) as demonstrated in FIG. 6Dand then applying the error function.

When both R and X signals are available from all antennae locationsH_(ll), H_(uu), and H_(tt), R_(H) and R_(V) may be determinedsimultaneously with greater accuracy by minimizing the error function:$\begin{matrix}\begin{matrix}{{error} = \quad \left| {H_{tt}^{measured} + {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {1 - u + u^{2}} \right\}}} \middle| {}_{2}\quad { +} \right.} \\{\quad {\left| \quad {H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \middle| {}_{2} + \right.}} \\{\quad \left| {H_{uu}^{measured} - {{\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {u\left( {1 - e^{u{({\alpha - 1})}}} \right)} +} \right.}}} \right.} \\\left. {\quad \left. {u^{2}\left( {1 - {\alpha \quad e^{u{({\beta - 1})}}}} \right)} \right\}} \right|^{2}\end{matrix} & \lbrack 48\rbrack\end{matrix}$

The two above methods address determination of R_(H) and R_(V) at themost extreme cases, i.e. θ=0° or 90°. More often than not, the deviationangle for the borehole will be somewhere within this range. The abovemethod for determining the formation vertical and horizontalresistivities may be used where the deviation angle is less than 30°.However the effect of the deviation angle becomes significant for valuesabove 30°. In which instance, the full form of Eqs. 19-24 must be used:$\quad {H_{ll} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}$$\quad {H_{tt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} + {u\frac{\cos^{2}\theta}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}$$\quad {H_{lt} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {u\frac{\cos \quad \theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}}$$H_{uu} = {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}$  H_(ul) = H_(ut) = 0

There are three variables that characterize the formation's anisotropiccharacteristics. H_(ll) depends only on u (hence, R_(H)) and β, whereas,H_(tt), H_(lt), and H_(uu) are all dependent on u, β, and θ. Hereinthere are four independent measurements, each having an R and Xcomponent, for this overly constrained model.

Using both the R and X signals from H_(ll) can be used to determine u(or R_(H)) and β, if both θ (recalling that β={square root over(cos²θ+α²sin²θ)} and as θ→90°, β→α) and R_(V)/R_(H) are large. Thisrelationship is demonstrated in nomograph FIG. 7A, from which one maydetermining R_(H) and β as a function of the R and X signals from H_(ll)for varying R_(H) and β. R_(H) and β may also be determined from the Rand X signals of H_(ll) by minimization of the error function:$\begin{matrix}{{error} = \left| {H_{ll} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \right|^{2}} & \lbrack 49\rbrack\end{matrix}$

Alternatively, R_(H) may be determined from the ratio of the R/X signalsat H_(ll) as shown in nomograph FIG. 7A. Upon determining R_(H) (or u)and β, the may be substituted into Eqs. 20-22 to determine the remainingtwo variables, R_(V) and θ. Thus, one can determine the horizontal andvertical resistivities, R_(H) and R_(V), without prior knowledge of thedeviation angle θ, which may be independently determined.

When R and X signals are available from H_(ll), H_(tt), H_(lt), andH_(uu), then R_(H) and R_(V) and θ may be determined by minimizing theerror function: $\begin{matrix}\begin{matrix}{{error} = \quad \left| {H_{ll}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \middle| {}_{2} + \right.} \\{\quad \left. {H_{tt}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} + {u\frac{\cos^{2}\theta}{\sin^{2}\quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}} \middle| {}_{2} + \right.} \\{\quad \left. {H_{lt}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {u\frac{\cos \quad \theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}} \middle| {}_{2} + \right.} \\{\quad {H_{uu}^{measured} - {\frac{M_{0}}{4\pi}\frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} +} \right.}}} \\\left. {\quad \left. {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)} \right\}} \right|^{2}\end{matrix} & \lbrack 50\rbrack\end{matrix}$

Commercially available computer programs may be used to minimize theerror function of Eq. 50. The minimization of the error as defined inEq. 50 permits the use of measured values, as opposed to values thathave been rotated through the azimuth and dip corrections, tosimultaneously determine R_(H), R_(V) and θ.

Note that if the deviation angle θ is already known, and is not small,then the any or all of the Eqs. 20-22 may be used to determine R_(H) andR_(V). For instance, both the R and X signals from H_(ll) can be used todetermine R_(H) and β, then R_(V) from β if angle θ is known.

FIG. 8 is a flow chart depicting the operation of the method of thepresent invention. In step 100, the orthogonal array induction tool 10is either located opposite the formation of interest 4 on a wireline 6or as part of an LWD tool (not shown). In step 102, the formationinversion model is created for the formation and the expected response.It will be appreciated that the sequence of steps between 100 and 102are not dependent upon each other. i.e., the inversion model may bedeveloped prior to positioning the induction tool opposite the formationof interest. In step 104, the induction tool 10 is energized to transmitan oscillating signal into the formation 4. The signals transmitted intothe formation 4 create eddy currents in the formation 4 which aremeasured up by the orthogonal receiver array in tool 10. In receivingthe signals, the tool retains both the resistive and reactive portion ofthe received signal.

It is recognized that not all signal values measured by the receiverantenna may be independent. As such, the azimuthal angle φ may bedetermined in step 108 based on the measured signals available. See Eqs.26-30. Based on the azimuthal angle and the measured signals, asecondary set of signals can be determined from the measured signals instep 110. See. Eq. 9.

Selecting from the available secondary signals, which themselves aregenerated from available measured signals in step 112. The presentinvention determines the horizontal and vertical resistivity, along withthe dip, simultaneously by minimization of error functions based on theinversion models in step 114. See, Eqs. 38-40 and 45-49. The specificminimization function in the equations is based on those secondarysignals available. In step 116 a signal representative of the verticaland horizontal resistivity, as well as the dip angle is generated andoutput by tool 10.

The signal may then be transmitted up to a surface system utilizing awireline 6 or by some other method from an LWD tool, such as pressurepulse or RF telemetry.

Accordingly, the preferred embodiment of the present invention disclosesa means for determining earth formation anisotropic resistivityutilizing a multi-component induction tool. Moreover, a method isdisclosed for performing inversion techniques to determine the saidcharacteristics utilizing various combinations of R and X signals atvarying antenna locations. Moreover, a method of determining formationdeviation or dip angle has been disclosed.

While the invention is susceptible to various modifications andalternative forms, specific embodiments have been shown by way ofexample in the drawings and that have been described in detail herein.However, it should be understood that the invention is not intended tobe limited to the particular forms disclosed. Rather, the invention isto cover all modifications, equivalents, an alternatives falling withinthe spirit and scope of the present invention as defined by the appendedclaims.

Appendix 1 Transformation between Formation and Borehole CoordinateSystems

Consider a multi-component induction resistivity tool whose antennadirections (1, m, n) are aligned to the (z, x, y) axis in the formation.$H^{formation} = \begin{bmatrix}H_{zz} & H_{zx} & H_{zy} \\H_{xz} & H_{xx} & H_{xy} \\H_{yz} & H_{yx} & H_{yy}\end{bmatrix}$

Consider a deviated borehole in the formation. The deviation angle isnoted as θ. Without loss of generality, the deviated borehole is in thex-axis direction horizontally (FIG. 2). Consider a borehole coordinatesystem (l, t, u) where the u-axis coincides with the formation's y-axis.The borehole coordinate system and the formation coordinate system arerelated by a rotational operation around the y-axis by an angle θ.$\begin{bmatrix}\hat{l} \\\hat{t} \\\hat{u}\end{bmatrix} = {{\begin{bmatrix}{\cos \quad \theta} & {\sin \quad \theta} & 0 \\{{- \sin}\quad \theta} & {\cos \quad \theta} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}\hat{z} \\\hat{x} \\\hat{y}\end{bmatrix}} = {{R_{y}(\theta)}\begin{bmatrix}\hat{z} \\\hat{x} \\\hat{y}\end{bmatrix}}}$

Consider the multi-component induction tool whose antenna directions arealigned to the (l, t, u) direction. $H^{borehole} = \begin{bmatrix}H_{ll} & H_{xy} & H_{xz} \\H_{yx} & H_{yy} & H_{yz} \\H_{zx} & H_{zy} & H_{zz}\end{bmatrix}$

H^(borehole) and H^(formation) are related by the rotation as,$\begin{matrix}{\begin{bmatrix}H_{zz} & H_{zx} & H_{zy} \\H_{xz} & H_{xx} & H_{xy} \\H_{yz} & H_{yx} & H_{yy}\end{bmatrix} = \quad {{{R_{y}(\theta)}^{tr}\begin{bmatrix}H_{ll} & H_{lt} & H_{lu} \\H_{tl} & H_{tt} & H_{tu} \\H_{ul} & H_{ut} & H_{uu}\end{bmatrix}}{R_{y}(\theta)}}} \\{= \quad {{\begin{bmatrix}{\cos \quad \theta} & {{- \sin}\quad \theta} & 0 \\{\sin \quad \theta} & {\cos \quad \theta} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}H_{ll} & H_{lt} & H_{lu} \\H_{tl} & H_{tt} & H_{tu} \\H_{ul} & H_{ut} & H_{uu}\end{bmatrix}}\begin{bmatrix}{\cos \quad \theta} & {\sin \quad \theta} & 0 \\{{- \sin}\quad \theta} & {\cos \quad \theta} & 0 \\0 & 0 & 1\end{bmatrix}}}\end{matrix}$

Or, in terms of individual components,$H_{zz} = {{{H_{ll}\cos^{2}\theta} + {H_{tt}\sin^{2}\theta} - {2\quad \cos \quad \theta \quad \sin \quad \theta \quad H_{lt}}} = {\frac{H_{ll} + H_{tt}}{2} + {\frac{H_{ll} - H_{tt}}{2}\cos \quad 2\quad \theta} - {H_{lt}\sin \quad 2\quad \theta}}}$$H_{xx} = {{{H_{ll}\sin^{2}\theta} + {H_{tt}\cos^{2}\theta} + {2\quad \cos \quad \theta \quad \sin \quad \theta \quad H_{lt}}} = {\frac{H_{ll} + H_{tt}}{2} - {\frac{H_{ll} - H_{tt}}{2}\cos \quad 2\quad \theta} + {H_{lt}\sin \quad 2\quad \theta}}}$$H_{zx} = {{{\left( {H_{ll} - H_{tt}} \right)\cos \quad \theta \quad \sin \quad \theta} + {\left( {{\cos^{2}\theta} - {\sin^{2}\theta}} \right)H_{lt}}} = {{\frac{H_{ll} - H_{tt}}{2}\sin \quad 2\quad \theta} + {H_{lt}\cos \quad 2\quad \theta}}}$

 H _(yz) =H _(ul) cosθ−H _(ul) sin θ

H _(yx) =H _(ul) sin θ+H _(ul cos θ)

H _(yy) =H _(uu)

The first three equations are rewritten as,

H _(xx) +H _(xx) =H _(ll) +H _(ll)$\frac{H_{zz} - H_{xx}}{2} = {{\frac{H_{ll} - H_{tt}}{2}\cos \quad 2\quad \theta} - {H_{lt}\sin \quad 2\quad \theta}}$$H_{zx} = {{\frac{H_{ll} - H_{tt}}{2}\sin \quad 2\quad \theta} + {H_{lt}\cos \quad 2\quad \theta}}$

There are three independent invariants under this rotation:

H _(yy) =H _(uu)

H _(xx) +H _(xx) =H _(ll) +H _(ll)${\left( \frac{H_{zz} - H_{xx}}{2} \right)^{2} + H_{zx}^{2}} = {\left( \frac{H_{ll} - H_{tt}}{2} \right)^{2} + H_{lt}^{2}}$

The right hand sides in all three equations can be related to themeasurements. The left-hand sides are expressed as shown below asfunctions of σH, σv, and θ. These three equations can be used to invertthe induction measurements to determine the formation anisotropicresistivity values and the deviation angle θ.

In a longitudinally anisotropic formation, H_(yz) and H_(yx) vanishes asshown below in Appendix 3. From this orthogonality condition,

H _(yz) =H _(ul) cos θ−H _(ul) sin θ=0

H _(yz) =H _(ul) sin θ+H _(ul) cos θ=0

the following two measurements are also zeros:

H _(ul) =H _(ul)=0.

Appendix 2 Transformation between Tool and Borehole Coordinate Systems

Rotate the tool around the tool's l-axis by an angle φ $\begin{bmatrix}\hat{l} \\\hat{m} \\\hat{n}\end{bmatrix} = {{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos \quad \phi} & {\sin \quad \phi} \\0 & {{- \sin}\quad \phi} & {\cos \quad \phi}\end{bmatrix}\begin{bmatrix}\hat{l} \\\hat{t} \\\hat{u}\end{bmatrix}} = {{R_{l}(\phi)}\begin{bmatrix}\hat{l} \\\hat{t} \\\hat{u}\end{bmatrix}}}$

H^(borehole) and H^(tool) are related by the rotation as,$\begin{matrix}{\begin{bmatrix}H_{ll} & H_{lt} & H_{lu} \\H_{tl} & H_{tt} & H_{tu} \\H_{ul} & H_{ut} & H_{uu}\end{bmatrix} = \quad {{{R_{l}(\phi)}^{tr}\begin{bmatrix}H_{ll} & H_{tm} & H_{\ln} \\H_{ml} & H_{mm} & H_{mn} \\H_{nl} & H_{nm} & H_{nn}\end{bmatrix}}{R_{l}(\phi)}}} \\{= \quad {{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos \quad \phi} & {\text{-}\sin \quad \phi} \\0 & {\sin \quad \phi} & {\cos \quad \phi}\end{bmatrix}\begin{bmatrix}H_{ll} & H_{lm} & H_{\ln} \\H_{ml} & H_{mm} & H_{mn} \\H_{nl} & H_{nm} & H_{nn}\end{bmatrix}}\begin{bmatrix}1 & 0 & 0 \\0 & {\cos \quad \phi} & {\sin \quad \phi} \\0 & {\text{-}\sin \quad \phi} & {\cos \quad \phi}\end{bmatrix}}}\end{matrix}$

Or, in terms of individual components,

H _(ll) =H _(ll)

H _(lt) =H _(lm) cos φ−H _(ln) sin φ

H _(lu) =H _(lm) sin φ+H _(ln) cos φ$H_{tt} = {{{H_{mm}\cos^{2}\phi} + {H_{nn}\sin^{2}\phi} - {2\quad \cos \quad \phi \quad \sin \quad \phi \quad H_{mn}}} = {\frac{H_{mm} + H_{nn}}{2} + {\frac{H_{mm} - H_{nn}}{2}\cos \quad 2\quad \phi} - {H_{mn}\sin \quad 2\phi}}}$$H_{uu} = {{{H_{mm}\sin^{2}\phi} + {H_{nn}\cos^{2}\phi} + {2\quad \cos \quad \phi \quad \sin \quad \phi \quad H_{mn}}} = {\frac{H_{mm} + H_{nn}}{2} - {\frac{H_{mm} - H_{nn}}{2}\cos \quad 2\quad \phi} + {H_{mn}\sin \quad 2\quad \phi}}}$$H_{tu} = {{{\left( {H_{mm} - H_{nn}} \right)\cos \quad \phi \quad \sin \quad \phi} + {\left( {{\cos^{2}\phi} - {\sin^{2}\phi}} \right)H_{mn}}} = {{\frac{H_{mm} - H_{nn}}{2}\sin \quad 2\quad \phi} + {H_{mn}\cos \quad 2\quad \phi}}}$

The last three equations are rewritten as,

H _(ll) +H _(uu) =H _(mm) +H _(nn)$\frac{H_{tt} - H_{uu}}{2} = {{\frac{H_{mm} - H_{nn}}{2}\cos \quad 2\quad \phi} - {H_{mn}\sin \quad 2\quad \phi}}$$H_{tu} = {{\frac{H_{mm} - H_{nn}}{2}\sin \quad 2\quad \phi} + {H_{mn}\cos \quad 2\quad \phi}}$

There are three independent invariants under this rotation.

H _(ll) =H _(ll)

H _(ll) +H _(uu) =H _(mm) =H _(nn)${\left( \frac{H_{tt} - H_{uu}}{2} \right)^{2} + H_{tu}^{2}} = {\left( \frac{H_{mm} - H_{nn}}{2} \right)^{2} + H_{mn}^{2}}$

The orthogonality condition, H_(ul=H) _(ul)=0, implies

H _(lu) =H _(lm) sin φ+H _(ln) cos φ=0$H_{tu} = {{{\frac{H_{mm} - H_{nn}}{2}\sin \quad 2\quad \phi} + {H_{mn}\cos \quad 2\quad \phi}} = 0}$

Thus, the azimuthal angle φ can be determined from a set of (H_(lm),H_(ln)) measurements or (H_(mm), H_(nn), H_(mn)) measurements.

is solved and the azimuthal angle α is given by,${{{\tan \quad \phi} = {- \quad \frac{H_{\ln}}{H_{lm}}}};\quad {{\cos \quad \phi} = \frac{H_{lm}}{\sqrt{H_{lm}^{2} + H_{\ln}^{2}}}}},\quad {{\sin \quad \phi} = {- \quad \frac{H_{\ln}}{\sqrt{H_{lm}^{2} + H_{\ln}^{2}}}}}$${\tan \quad 2\quad \phi} = {{- \quad \frac{2H_{mn}}{H_{mm} - H_{nn}}} = \frac{2\quad \tan \quad \phi}{1 - {\tan^{2}\phi}}}$

And, (H_(lm), H_(ln), H_(mm), H_(nn), H_(mn)) must satisfy the followingconsistency condition:$H_{mn} = \frac{\left( {H_{mm} - H_{nn}} \right)H_{lm}H_{\ln}}{\left( {H_{lm}^{2} - H_{\ln}^{2}} \right)}$

This is obtained by eliminating φ as $\begin{matrix}{H_{tu} = \quad {{\frac{H_{mm} - H_{nn}}{2}\sin \quad 2\quad \phi} + {H_{mn}\cos \quad 2\quad \phi}}} \\{= \quad {\frac{{{- \left( {H_{mm} - H_{nn}} \right)}H_{lm}H_{\ln}} + {\left( {H_{lm}^{2} - H_{\ln}^{2}} \right)H_{mn}}}{H_{lm}^{2} + H_{\ln}^{2}} = 0}}\end{matrix}$

The four measurements (H_(ll), H_(lt), H_(tt), H_(uu)) in the borehole(l,t,u) coordinate system are provided with five (H_(ll), H_(lm),H_(ln), H_(mm), H_(nn)) measurements in the (l,m,n) coordinate system asbelow:

H _(ll) =H _(ll) $\begin{matrix}{H_{lt} = \quad {{{H_{lm}\cos \quad \phi} - {H_{\ln}\sin \quad \phi}} = \sqrt{H_{lm}^{2} + H_{\ln}^{2}}}} \\{H_{tt} = \quad {{H_{mm}\cos^{2}\phi} + {H_{nn}\sin^{2}\phi} - {2\quad \cos \quad \phi \quad \sin \quad \phi \quad H_{mn}}}} \\{= \quad {\frac{H_{mm} + H_{nn}}{2} + {\frac{H_{mm} - H_{nn}}{2}\cos \quad 2\quad \phi} - {H_{mn}\sin \quad 2\quad \phi}}} \\{= \quad \frac{{H_{mm}H_{lm}^{2}} + {H_{nm}H_{\ln}^{2}} + {2H_{lm}H_{\ln}H_{mn}}}{H_{lm}^{2} + H_{\ln}^{2}}} \\{= \quad \frac{{H_{mm}H_{lm}^{2}} - {H_{nn}H_{\ln}^{2}}}{H_{lm}^{2} - H_{\ln}^{2}}} \\{H_{uu} = \quad {{H_{mm}\sin^{2}\phi} + {H_{nn}\cos^{2}\phi} + {2\quad \cos \quad \phi \quad \sin \quad \phi \quad H_{mn}}}} \\{= \quad {\frac{H_{mm} + H_{nn}}{2} - {\frac{H_{mm} - H_{nn}}{2}\cos \quad 2\quad \phi} + {H_{mn}\sin \quad 2\quad \phi}}} \\{= \quad \frac{{H_{mm}H_{\ln}^{2}} + {H_{nn}H_{lm}^{2}} - {2\quad H_{lm}H_{\ln}H_{mn}}}{H_{lm}^{2} + H_{\ln}^{2}}} \\{= \quad \frac{{H_{nn}H_{lm}^{2}} - {H_{mm}H_{\ln}^{2}}}{H_{lm}^{2} - H_{\ln}^{2}}}\end{matrix}$

If H_(mn) is measured additionally, then the orthogonality conditionabove can be used for a consistency check (QC).

Appendix 3 Determination of σH, σv, and θ.

In the formation coordinate system (z, x, y),$H_{zz} = {\frac{M_{z}}{4\quad \pi}\left\{ {{\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\quad \frac{z^{2}}{r^{2}}} - \left( {1 - u + u^{2}} \right)} \right\} \quad \frac{e^{u}}{r^{3}}}$$H_{zx} = {\frac{M_{z}}{4\quad \pi}\left\{ {\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\quad \frac{xz}{r^{2}}} \right\} \quad \frac{e^{u}}{r^{3}}}$$H_{xx} = {{\frac{M_{x}}{4\quad \pi}\left\{ {{\left( {{3\left( {1 - u} \right)} + u^{2}} \right)\quad \frac{x^{2}}{r^{2}}} - \left( {1 - u + u^{2}} \right)} \right\} \quad \frac{e^{u}}{r^{3}}} + I_{0}}$$H_{yy} = {{\frac{M_{y}}{4\quad \pi}\left\{ {- \left( {1 - u + u^{2}} \right)} \right\} \quad \frac{e^{u}}{r^{3}}} - I_{0} + I_{1}}$

where

u=ik _(H) r

r={square root over (x²+y²+z²)}; ρ={square root over ( x ² +y ²)}

and at a receiver located at (z,x,y)=(Lcosθ, Lsinθ, 0), $\begin{matrix}{I_{0} = \quad {{\frac{M_{z}}{4\quad \pi}\left\{ {\frac{u}{r}\left( {e^{u} - e^{u\quad \beta}} \right)} \right\} \quad \frac{1}{\rho^{2}}} = {\frac{M_{z}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}}} \\{= \quad {\frac{M_{z}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}{\overset{\sim}{I}}_{0}}} \\{I_{1} = \quad {{\frac{M_{z}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}{u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} = {\frac{M_{z}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}{\overset{\sim}{I}}_{1}}}} \\{{\beta = \quad \sqrt{{\cos^{2}\theta} + {\alpha^{2}\sin^{2}\theta}}};\quad {\alpha^{2} = \frac{\sigma_{V}}{\sigma_{H}}}}\end{matrix}$

 r=L; ρ=Lsin θ

We also set that all the transmitter have the identical strength:M_(l)=M_(t)=M_(u)=M₀.

Then,$\quad {{H_{zz} + H_{xx}} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {\left\lbrack {1 - u - u^{2}} \right\rbrack + {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}$${H_{zz} - H_{xx}} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{\left\lbrack {{3\left( {1 - u} \right)} + u^{2}} \right\rbrack \left( {{\cos^{2}\theta} - {\sin^{2}\theta}} \right)} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}$$\quad {H_{zx} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{3\left( {1 - u} \right)} + u^{2}} \right\} \cos \quad \theta \quad \sin \quad \theta}}$$H_{yy} = {H_{yy} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}}$

Three invariants are then written as,$\quad {{H_{zz} + H_{xx}} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {\left\lbrack {1 - u - u^{2}} \right\rbrack + {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}$${\left( \frac{H_{zz} - H_{xx}}{2} \right)^{2} + H_{zx}^{2}} = {\frac{1}{4}\left( {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}} \right)^{2}\left\{ {\left( {{3\left( {1 - u} \right)} + u^{2}} \right)^{2} - {2\quad {\overset{\sim}{I}}_{0}\cos \quad 2\quad {\theta \left\lbrack {{3\left( {1 - u} \right)} + u^{2}} \right\rbrack}} + {\overset{\sim}{I}}_{0}^{2}} \right\}}$$H_{yy} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}$

where${\overset{\sim}{I}}_{0} = {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}$${\beta = \sqrt{{\cos^{2}\theta} + {\alpha^{2}\sin^{2}\theta}}};\quad {\alpha^{2} = {\frac{1}{\lambda^{2}} = \frac{\sigma_{V}}{\sigma_{H}}}}$

There are three invariants: There are three unknowns, k_(H), λ, and θ.

Or, simply using all 4 equations for H_(xx), H_(xz), H_(zz), and H_(yy),one can determine k_(H), λ, and θ.

Since

H _(yy) =H _(uu)

H _(zz) +H _(xx) =H _(ll) +H _(ll)${\left( \frac{H_{zz} - H_{xx}}{2} \right)^{2} + H_{zx}^{2}} = {\left( \frac{H_{ll} - H_{tt}}{2} \right)^{2} + H_{lt}^{2}}$

the above equations hold also in the (l,t,u) tool coordinates.

But, it turns out the following may be easier to solve for k_(H), β, andθ, hence λ.$\quad {H_{ll} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}$$\quad {H_{tt} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{- {1\left\lbrack {1 - u + u^{2}} \right\rbrack}} + {u\quad \frac{\cos^{2}\theta}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}}$$\quad {H_{lt} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {u\quad \frac{\cos \quad \theta}{\sin \quad \theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} \right\}}}$$H_{uu} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {\frac{u}{\sin^{2}\theta}\left( {1 - e^{u{({\beta - 1})}}} \right)} + {u^{2}\left( {1 - {\frac{\alpha^{2}}{\beta}e^{u{({\beta - 1})}}}} \right)}} \right\}}$

At θ=0,$H_{ll} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {2\left\lbrack {1 - u} \right\rbrack} \right\}}$$H_{tt} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {- \left\lbrack {1 - u + {u^{2}\quad \frac{\alpha^{2} + 1}{2}}} \right\rbrack} \right\}}$

 H _(ll)=0

H _(uu) =H _(tt)

At θ=/2,$H_{ll} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{2\left\lbrack {1 - u} \right\rbrack} + {u\left( {1 - e^{u{({\beta - 1})}}} \right)}} \right\}}$$H_{tt} = {{- \quad \frac{M_{0}}{4\quad \pi}}\quad \frac{e^{u}}{r^{3}}\left\{ {1 - u + u^{2}} \right\}}$$H_{uu} = {\frac{M_{0}}{4\quad \pi}\quad \frac{e^{u}}{r^{3}}\left\{ {{- \left\lbrack {1 - u + u^{2}} \right\rbrack} - {u\left( {1 - e^{u{({\alpha - 1})}}} \right)} + {u^{2}\left( {1 - {\alpha \quad e^{u{({\beta - 1})}}}} \right)}} \right\}}$

 H _(lt)=0

H _(ul)=0

H _(ut)=0

What is claimed:
 1. A method for determining the horizontal resistivityand vertical resistivity of an earth formation, the earth formationbeing penetrated by a borehole, comprising: (a) developing an inversionmodel for various earth formations; (b) deploying an induction tool insaid borehole, said tool having a longitudinal axis, a transmitter arraycomprised three mutually orthogonal transmitter antennae, at least oneof the antenna being oriented parallel to said tool longitudinal axis,and a receiver array offset from the transmitter array, said receiverarray being comprised of three mutually orthogonal receiver antennae,the receiving array sharing a common orientation with said transmitterarray; (c) activating said transmitter antenna array and measuringelectromagnetic signals values induced in said receiving array antennae,including resistive and reactive components of said signal values; (d)determining an azimuth angle for said tool; (e) calculating secondarysignal values signals as a function of said measured signal values andsaid azimuth angle; and (f) simultaneously determining said horizontalresistivity, vertical resistivity and a dip angle as a function ofselected resistive and reactive components of said secondary signals byminimizing error utilizing said inversion model.
 2. The method of claim1, wherein said azimuth angle is calculated as a function of saidelectromagnetic signals in said receiver array antennae perpendicular tosaid tool longitudinal axis, relative to a direction of said horizontalresistivity and said vertical resistivity.
 3. The method of claim 1,wherein step (f) further includes the simultaneous determination of adip angle.
 4. The method of claim 1, wherein said secondary signalvalues are calculated by rotating said measured signals values through anegative of said azimuth angle.
 5. The method of claim 2, wherein step(f) further includes the simultaneous determination of a dip angle. 6.The method of claim 3, wherein said dip angle is calculated as afunction of said secondary values for said electromagnetic signals for areceiver antenna parallel to said tool longitudinal axis with respect tosaid vertical resistivity.